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セミパラメトリック推論の基礎の復習
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Daisuke Yoneoka
November 14, 2023
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セミパラメトリック推論の基礎の復習
Daisuke Yoneoka
November 14, 2023
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Transcript
ηϛύϥϝτϦοΫਪͷجૅͷ෮श Daisuke Yoneoka September 29, 2014
Notations جຊతʹ Tsiatis,2006 ʹै͏. Θ͔Μͳ͔ͬͨΒࣗͰௐͯͶ! ϕΫτϧߦྻଠࣈʹͯ͠ͳ͍͚Ͳ, ͦࣗ͜Ͱิ͍ͬͯͩ͘͞. σʔλ i.i.d Ͱ
Zi = (Zi1, . . . , Zim) ∈ Rm αϯϓϧαΠζ n ਓ. i.e., Z1, . . . , Zn φ(Z) Өڹؔ u(Zi, θ) ਪఆؔ Լ͖ࣈͷ eff (ۙ) ༗ޮ (efficient) ͱ͍͏ҙຯ
ηϛύϥϝτϦοΫਪͱʁ Zi ͷີ͕ؔηϛύϥϝτϦοΫϞσϧʹै͏ͱ S = {p(z : θ, η)|θ ∈
Θ ⊂ Rr, η ∈ H} θ ༗ݶ࣍ݩͷڵຯ͋ΔύϥϝλͰ, η ແݶ࣍ݩͷͲ͏Ͱ͍͍ύ ϥϝλ (ہ֎ (nuisance) ύϥϝʔλʔ). ηϛύϥϝτϦοΫਪ: ͜ͷͱͰ θ ͷ࠷ྑͷਪఆྔ (RAL ਪఆ ྔ) ΛͱΊΔ͜ͱ
Өڹؔ θ ͳΜͰ͍͍͔Β࠷ྑΛݟ͚ͭΔͱ͍͏ͷແཧήʔ → Ϋϥε Λݶఆͯͦ͜͠Ͱݟ͚ͭΔ! (౷ܭͰΑ͘ΔΑͶ) Өڹؔ: ਪఆྔ ˆ
θ ͷӨڹؔͱ, (Ϟʔϝϯτʹ੍͕͋Δ) √ n(ˆ θ − θ) = 1 √ n n i=1 φ(Zi, θ, η) + op(1) Λຬͨ͢ϕΫτϧؔ. ˆ θ ۙઢܗਪఆྔͱݺͼ n → ∞ ͰҰகੑ ͱۙਖ਼نੑ͕͋Δ √ n(ˆ θ − θ) → N 0, E[φ(Zi, θ, η)φ(Zi, θ, η)T ] Πϝʔδతʹ͋Δσʔλ͕ͲΕ͚ͩਪఆʹӨڹΛ༩͍͑ͯΔ͔Λ දݱͨ͠ͷ
ਪఆؔͱ M ਪఆ ਪఆํఔࣜ n i=1 u(Zi, θ) ਪఆؔ =
0 ͷղͱͯ͠ಘΒΕΔͷΛ M ਪఆྔ ͱݺͿ. Α͘ݟΔ score ؔͳΜ͔ίϨ. ͨͩ͠, E[φ(Zi, θ)] = 0 ظ 0 , E[∥φ(Zi, θ)∥2] < ∞ ࢄతͳͷൃࢄ͠ͳ͍ . ͋ͱ͏গ͚ͩ݅͋͠Δ. Ұகੑͱۙਖ਼نੑΛ࣋ͭ √ n(ˆ θ − θ) = 1 √ n n i=1 E[ ∂u(Zi, θ) ∂θ ] −1 u(Zi, θ) ͕͜͜Өڹؔʹͳ͍ͬͯΔ +op(1) → N 0, E[ ∂u(Zi, θ) ∂θ ] −1 E[u(Zi, θ)u(Zi, θ)T ] E[ ∂u(Zi, θ) ∂θ ] −T ] ͜ͷۙࢄͷਪఆྔΛαϯυΠονਪఆྔͱݺΜͩΓ͢Δ
RAL ਪఆྔ ۙઢܥਪఆྔͳΜ͔ྑͦ͞͏ʂͰ super efficiency ͷ (Hodges) ͕Δʂ Super efficiency:
ۙతʹ Cramer-Rao ͷԼݶΑΓྑ͍ͷ͕Ͱ͖ Δͷ͜ͱ ͜ͷΛղܾͨ͠ͷ͕ RAL (Regular asymptotic linear) ਪఆྔ. ͦͷਖ਼ଇ݅ۃݶ͕ LDGP (local data generating process) ʹґ ଘ͠ͳ͍͜ͱ (ৄ͘͠ Tsiatis, 2006) ηϛύϥਪ͜ͷ RAL ਪఆྔͷӨڹؔΛٻΊΔ͜ͱΛߟ͑Δ
Parametric submodel ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠ p(z; θ, η) ∈ Ssub
⊂ S Λຬͨ͢ύϥϝτϦοΫϞσϧ Ssub = {p(z; θ, γ)|θ ∈ Θ ⊂ Rr, γ ∈ Γ ⊂ Rs, s ∈ N} ΛύϥϝτϦοΫαϒϞσϧͱݺͿ.
Nuisance tangent space (ہ֎ۭؒ) ηϛύϥϝτϦοΫϞσϧ S ͷ֤ʹର͠, ύϥϝτϦοΫαϒϞσϧ Ssub ͷہ֎ۭؒΛ
TN θ,γ (Ssub) = {BT sγ(z, θ, γ)|B ∈ Rs} ͱ͢Δ. γ p(z; θ, η) ʹରԠ͢ΔͷͰ sγ(z, θ, γ) = ∂ ∂γ log p(z; θ, γ) Ͱ ද͞ΕΔ nuisance score ؔ. ͜ͷઢܗۭؒ͜ͷ nuisance score vector ʹ ΑͬͯுΒΕ͍ͯΔ. ͜ͷͱ͖ TN θ,η (S) = Ssub TN θ,γ (Ssub) Λ S ্ͷ p(z; θ, η) ʹ͓͚Δہ֎ۭؒͱΑͿ. ͪͳΈʹ, ଆͷू ߹ʹؔͯ͠ closure ΛͱΔԋࢉࢠ. Note:͜ͷۭؒେͰޙʹ, RAL ਪఆྔͷӨڹؔ͜ͷۭؒʹަۭͨؒ͠ʹ ଐ͢Δ͜ͱ͕ॏཁʹͳͬͯ͘Δʂ
ઢܗ෦ۭؒͷࣹӨͷزԿͱϐλΰϥεͷఆཧ
RAL ਪఆྔͷӨڹؔͷॏཁͳఆཧ ηϛύϥϝτϦοΫ RAL ਪఆྔ β ͷӨڹؔ φ(Z) ҎԼͷ݅Λຬ ͢Δ.
Corollary1 E[φ(Z)sβ] = E[φ(Z)sT efficient (Z, β0, η0)] = I. ͨͩ͠, s είΞؔͰ, sT efficient ༗ޮείΞؔ Corollary2 φ(Z) ہ֎ۭؒʹަ͍ͯ͠Δ. ༗ޮӨڹ্ؔͷ 2 ͭͷ݅Λຬͨ͠, ͦͷࢄߦྻ, ޮݶքΛୡ ͦ͠Ε φeffi(Z, β0, η0) = E[seff (Z, β0, η0)sT eff (Z, β0, η0)] −1 seff (Z, β0, η0)
ηϛύϥۭؒͷఆཧ ύϥϝτϦοΫαϒϞσϧͷ߹ͷ RAL ਪఆྔͷӨڹؔͱۭؒͱͷؔ Tsiatis, 2006 ͷ Ch4.3 ͋ͨΓΛݟͯͶʂ ఆཧ
1 RAL ਪఆྔͷӨڹؔ {φ(Z) + TN θ,η (S)⊥} ͱ͍͏ۭؒʹؚ·ΕΔ. ͨͩ͠, φ(Z) ҙͷ RAL ਪఆྔͷӨڹؔͰ, TN θ,η (S)⊥ ηϛύϥϝτϦο Ϋۭؒͷަิۭؒ ఆཧ 2 ηϛύϥϝτϦοΫ༗ޮͳਪఆྔ, ͦͷӨڹ͕ؔҰҙʹ well-defined Ͱܾఆ͞ Ε,φefficient = φ(Z) − {φ(Z)|TN θ,η (S)⊥} ͷཁૉ. ͪͳΈʹ, (h|U) projection of h ∈ H(ੵΛಋೖͨ͠ώϧϕϧτۭؒ) onto the space U (ઢܗۭؒ)
GEE ʹ͍ͭͯͷ Remarks Liang-Zeger ͷ GEE ͷηϛύϥϝτϦοΫϞσϧ (੍ϞʔϝϯτϞσϧ: 1 ࣍ͱ
2 ࣍ͷϞʔϝϯτʹ੍͚ͩΛஔ͍ͨϞσϧ) ҎԼͷಛΛͭ. ہॴ (ۙ༗) ޮਪఆྔ: ࢄؔͷԾఆ͕ਖ਼͚͠Ε, ༗ޮਪఆྔ Robustness: ແݶ࣍ݩͷύϥϝʔλਪఆ͕ඞཁ͕ͩ, ࢄؔΛ misspecify ͨ͠ͱͯ͠Ұகੑͱۙਖ਼نੑอ࣋ GEE ͷຊΛಡΊΘ͔Δ͚Ͳ, Working covariance matrix Λؒҧ͑ͯ ༗ޮੑࣦΘΕΔ͕, ͦͷଞͷ·͍͠ੑ࣭ (ۙਖ਼نੑͱҰகੑ) อ࣋Ͱ͖Δͬͯ͜ͱ